Optica Applicata, Vol. XXXIII, No. 2-3, 2003
Transformation of polarization state of the light
using wave plates with arbitrary phase difference:
half wave plates
Władysław A. Woźniak, Florian Ratajczyk
Institute o f Physics, Wroclaw University o f Technology, Wybrzeże Wyspiańskiego 2 7 ,5 0 -3 7 0 Wroclaw, Poland, e-mail: wladyslaw.woznak@pwr.wroc.pl.
The practical problem o f the lack o f a quarter-wave plate for an arbitrary wavelength has been solved in our earlier work. The possibility o f substituting this plate with a combination o f two (or more) retardation plates with phase shift different from 90° has been shown. However, there are some special cases in which another wave plate commonly used in polarization optics is needed,
i.e., a half-wave plate. In the present paper some formulae have been derived for several o f the
most important applications o f these retardation plates in measuring setup.
Keywords: polarized light, polarization, wave plate, phase difference.
1. Introduction
The synthesis and analysis of light with arbitrary polarization state is one o f the most important problems o f polarization optics. In all measuring methods, in which polarized light is used, a generation o f precisely controlled polarization state of the light is necessary as well as transformation o f the analyzed light - changes o f azimuth and ellipticity angles [1].
The basic tools used in polarization optics are always the same - a polarizer (in practice, a linear one) and two birefringent plates: one with phase difference between first and second eigenwaves equal to 90° (A/4 in wavelength) - a quarter-wave plate, and second with phase difference between first and second eigenwaves equal to 180°
(XI2 in wavelength) - a half-wave plate. A half-wave plate is commonly used in some
special cases and some most important of them will be discussed in this paper, e.g.:
- how to change the azimuth angle o f linearly polarized light;
- how to change the sign o f the ellipticity of light, without changing its azimuth angle;
- how to transform the light with a given polarization state into the light with orthogonal polarization state.
Fig. 1. Three most important cases o f using the half-wave plates in optical setups: change o f the azimuth angle o f linearly polarized light (a); change o f the sign o f the ellipticity o f light (b); change o f the polarization state o f light into perpendicular one (c).
The half-wave plate is easy to use for those purposes if only its phase difference is equal exactly to 180°. Figure 1 shows all the above-mentioned cases of using half -wave plate. The mismatch of phase shifts causes mismatch in parameters of the final polarization state o f light which could be treated as errors in many cases, but sometimes the difference between an “ideal” half-wave plate and the one which is at our disposal, is too big and could not be treated as an “error” . In a modern optical laboratory, a setup of good quality quarter- and half-wave plates for different wavelengths might be necessary, which is usually expensive and hard to achieve. As was shown in our earlier paper [2], due to the optical dispersion o f birefringence of materials used to make retardation plates, the difference in phase shift in wave-plate made for He-Ne laser light (632.8 nm) used with light of sodium lamp (589.3 nm) reaches about 10°. This means that such a phase plate will be practically unacceptable to use as a half-wave plate (in the meaning described above). We propose a way of solving this problem by using two (or more) phase plates with an arbitrary phase difference. In our further considerations, similarly as in [2], we decided to use the Poincare sphere formalism [3], [4], which let us find the parameters o f polarized light in a more intuitive way than solving a system o f complicated matrix equations.
Transformation o f polarization state o f the light using wave plates ... 339
2. Transformation of the azimuth angle of linearly polarized light
In this section, we analyze how to change the azimuth angle of linearly polarized light having at least two wave plates whose phase shifts are different from 180°. The polarization state of the input light is represented by point Sp with the azimuth angle ap (Fig. 2). The first plate (with the azimuth angle cq and phase difference y{)
transforms this state to the point S' (with the azimuth angle a' and ellipticity angle
while the second plate W2 (with the azimuth angle cq and phase difference y2) to the
point Sk (with the azimuth angle a k) on the equator of the sphere.
Based on the description given in Fig. 2, the following formulae for spherical triangle W^S'A could be easily written:
sin Wj 5"sin < = sinAS', (la )
sinW ^A tan^S'W jA = tan AS', (2a)
tan w js 'c o s < S'W XA = tan W^A (3a) where W{S' = W{Sp = 2(cq - ap), AS' = 2 6 \ W{A = 2 ( a ' - a l), <S'WYA = 180°- f t .
The same form ulae for spherical triangle W2S'A are:
sin W2S 's in < S 'W 2A = sinAS', (lb )
sin W2A tan <S'W2A = tan A S', (2b)
tan W2S'cos <S'W2A = tanW2A (3b)
where W2S' = W2Sk = 2(02 - a k), W2A = 2( o ' - cq), <S'W2A = 180° - y2.
In a general case, when y1 * y2, the system of equations obtained from spherical
geometry analysis could not be solved in an analytical way and desired quantities a x
and a 2 (azimuth angles o f two retardation plates) could be obtained using numerical
calculations. The analytical solution could be easily obtained when y{ = y2 = y, which
fortunately happens m ost commonly in optical laboratory (the wave plates, which have the same retardations for one wavelength, have also the same retardation for another wavelength). In this case, one can immediately obtain from Eqs. (la ), (lb ) and Eqs. (2a), (2b):
2 (4)
(the “interm ediate” polarization state S' lies exactly in the middle o f the arc Wx W2 on
the Poincare sphere), and
< * l - a p = a k - a 2 ( 5 )
(the position o f the first retardation plate Wx in relation to input state o f the light Sp
and the position of the second retardation plate W2 in relation to desired output state
o f the light Sk are the same).
Using the set of equations described above one can obtain the following quadratic equation:
(TkpC) X2 + ( C + l ) X - T kp = 0 (6)
where = tan(ak - ap\ C = cos(180° - y) (known values) and X = tan2(ce! - ap)
(calculated value). One can easily obtain from this equation the desired quantity ctj - ap as well as the conditions for the existence of possible solutions for this quantity.
The question is: which solution could be chosen (two possible solutions o f quadratic equations and periodicity o f tangent function) but they are common for any polariza tion calculations and the authors of the present paper are convinced that detailed explanations are unnecessary.
3. Transformation of the sign of the eliipticity of light,
without changing its azimuth angle
This section is devoted to the problem of how to change the rotation o f the light polarization state, that is the sign of the eliipticity of light without changing its azimuth angle. For retardation plate with phase difference y equal exactly to 180° the solution is simple and intuitive, as shown in Fig. lb . For plates with phase difference ydifferent from 180° the problem is also easy to solve using the Poincare sphere formalism and Fig. 3 shows the right way. The polarization state of the input light is represented by point Sp with the azimuth angle ap. The retardation plate W (with the azimuth angle a
Transformation ofpolarization state o f the light using wave plates ... 341
Fig. 3. Transformation o f the ellipticity sign o f light.
and phase difference y) transforms this state to the point Sk (with the azimuth angle ak). The following formulae for spherical triangle WSpA could be written:
sin WAtan <SpWA = tanASp (7)
where ASp = 2dp, WA = 2(ap - a), <SpWA = y/2. Note that in this case, only one
retardation phase plate is needed, and its position according to the azimuth angle of input light could be easily calculated from Eq. (7)
s in 2 ( a p - a ) = tan 2 #
tan y / 2 ' (8)
Moreover, the conditions for the existence of possible solutions for the azimuth angle ap are also immediately obtained from Eq. (7)
| tan 2 < (9)
which means that only for retardation plate with phase difference y< 180° all possible polarization states could be transformed into states with opposite ellipticity.
4. Transformation of the polarization state of light
into the orthogonal one
In this case, the system of equations, obtained from spherical geometry analysis is also not easy to solve in analytical way even if yl = y2 = y. There is no symmetry in general
case due to the fact that retardation plates are linear, and the points W] and W2 which
represent first eigenvectors of these plates always should be placed on the equator of the sphere (Fig. 4). However, another simplifying assumption could be made that the transformation from the point Sp (which represents the input light with azimuth angle a p and ellipticity angle &p) to the point Sk (which represents the output light with
Fig. 5. Transformation o f the polarization state of light into the orthogonal one using two wave plates with the same retardations y, = y2 =
Y-azimuth angle a k and ellipticity angle &k, note that a k = ap + 90° and &k = - due to the fact that Sk is perpendicular to Sp) is made through the intermediate point S' (with
the azimuth angle cO, which lies on the equator of the Poincare sphere. Due to this limiting assumption we can easily obtain the formulae for azimuth angles cq and cq of retardation plates and W2, respectively. We take the advantage of our earlier
paper [2] in which such situations as transformation from elliptically polarized light into linear one as well as the opposite transformation were described. The scheme of computation is shown in Fig. 5. Also in this case we have made our calculations for two identical retardation plates, i.e. yl = y2 = y (see the comments in Sec. 2). The
symmetry of the problem makes our computations easier. It readily follows from Fig. 5 that in this case the distance (on the Poincare sphere) between the first retardation plate
and polarization state of the input light Sp is equal to the distance between the
second plate W2 and output light Sk as shown by formula (5)
« ! - « , > = (>°)
Let us note that cq denotes the azimuth angle of the second eigenvector o f second retardation plate W2, which simplifes our considerations. The following formula could
be written for spherical triangle WxSpA:
sin wJa tan < Sp W {A = tan ASp (11) where: ASp = 2$p, WXA = 2(cq - ap), <SpW{A = 180° - / (note that this is the same
formula as in Eq. (7)). The position of the first retardation plate Wx according to the
azimuth angle of input light could be easily calculated from Eq. (11)
sin2(cq - a p) tan 2fl
tan (180° - y)
(12)
Transformation o f polarization state o f the light using wave plates ... 343
5. Conclusions
The main objective of this paper consisted in solving the practical problem, which could arise in optics laboratory, namely, lack of a half-wave plate for an arbitrary wavelength. It has been shown how to make use of retardation plates with phase shift different from 180° in some special cases, in which half-wave plates are usually used. Sometimes, it appears necessary to use more than one plate, as shown in Sections 2 and 4. All the formulae obtained are valid for arbitrary phase differences of the component plates and may be useful in some cases, especially if these differences are close to 180° (due to the solving conditions).
References
[1] Brosseau C., Fundamentals o f Polarized Light: a Statistical Optics Approach, A Wiley-Intersci. Pub.
John Wiley & Sons, Inc., New York 1998.
[2] Ratajczyk F., Woźniak W.A., Kurzynowski P., Opt. Commun. 183 (2000), 1. [3] Poincarź H., Theorie Mathematique de la Lumiere, Georges Carre, Paris 1892, Vol. 2.
[4] Ratajczyk F., Kurzynowski P., Opt. Appl. 27 (1997), 255.
Received June 20, 2002 in revised form 15 October, 2002
